Public transportation in Beirut is definitely lacking. The best way to get around Beirut, and Lebanon as a whole, is to drive. Car rentals can be pretty expensive, but it’s somewhat offset by the cheap cost of fuel.
Your biggest threat is going to come from other drivers, who are by no means sticklers for the rules of the road. Traffic laws are not enforced and parking is an exercise in patience. The quality of the roads is also poor, with unexpected potholes and unending road construction. Driving in the mountains can be especially hazardous and should be avoided.
If you’re not experienced in driving in crazy conditions, stick to cabs.